Introduc corporate finance ch4

4.1 The One-Period Case: Net Present Valuepresent value of the expected cash flows, less the cost of the investment.. 4.2 The Multiperiod Case: Future Valueinvestment over many periods c

Net Present Value

Chapter Outline

4.1 The One-Period Case4.2 The Multiperiod Case4.3 Compounding Periods4.4 Simplifications

4.5 What Is a Firm Worth?

4.6 Summary and Conclusions

• The late Sidney Homer, author of the classic, A History of Interest Rates, said that $1,000

invested at 8% for 400 years would grow to $23 quadrillion - $5 million for every human on

earth

What invariably happens is that someone with access to the money loses patience - the money burns a hole in his pocket

4.1 The One-Period Case: Future Value

one year, your investment would grow to $10,500

$500 would be interest ($10,000 × 05)

$10,000 is the principal repayment ($10,000 × 1)

$10,500 is the total due It can be calculated as:

$10,500 = $10,000×(1.05)

The total amount due at the end of the investment is call

the Future Value (FV)

4.1 The One-Period Case: Future Value

written as:

FV = C0×(1 + r) T

r is the appropriate interest rate.

4.1 The One-Period Case: Present Value

when interest rates are at 5-percent, your investment

be worth $9,523.81 in today’s dollars

05 1

000 ,

10

$ 81

523 ,

4.1 The One-Period Case: Present Value

written as:

r

C PV

+

=

1

1

r is the appropriate interest rate.

4.1 The One-Period Case: Net Present Value

present value of the expected cash flows, less the cost

of the investment

in one year is offered for sale for $9,500 Your interest rate is 5% Should you buy?

81 23

$

81 523 ,

9

$ 500

, 9

$

05 1

000 ,

10

$ 500

, 9

Yes!

4.1 The One-Period Case: Net Present Value

In the one-period case, the formula for NPV can be

written as:

NPV = –Cost + PV

If we had not undertaken the positive NPV project

considered on the last slide, and instead invested our

$9,500 elsewhere at 5-percent, our FV would be less

than the $10,000 the investment promised and we

would be unambiguously worse off in FV terms as

well:

$9,500×(1.05) = $9,975 < $10,000

4.2 The Multiperiod Case: Future Value

investment over many periods can be written as:

FV = C0×(1 + r) T

Where

r is the appropriate interest rate, and

T is the number of periods over which the cash is

invested

4.2 The Multiperiod Case: Future Value

offering of the Modigliani company Modigliani pays a current dividend of $1.10, which is expected

to grow at 40-percent per year for the next five years

FV = C0×(1 + r) T

Future Value and Compounding

considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend:

$5.92 > $1.10 + 5×[$1.10×.40] = $3.30 This is due to compounding.

Future Value and Compounding

10 1

$

3

) 40 1 ( 10 1

02 3

$

) 40 1 ( 10 1

54 1

$

2) 40 1 ( 10 1

16

2

$

5

) 40 1 ( 10 1

92 5

$

4

) 40 1 ( 10 1

23 4

$

Present Value and Compounding

today in order to have $20,000 five years from now

if the current rate is 15%?

000 ,

20

$ 53

943 ,

9

How Long is the Wait?

If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?

T

r C

FV = 0 × ( 1 + ) $ 10 , 000 = $ 5 , 000 × ( 1 10 )T

2 000

, 5

$

000 ,

10

$ )

10 1

2 ln )

10 1

years 27

.

7 0953

0

6931

0 )

10 1 ln(

Assume the total cost of a college education will be

$50,000 when your child enters college in 12 years You have $5,000 to invest today What rate of interest must you earn on your investment to cover the cost of your child’s education?

What Rate Is Enough?

T

r C

FV = 0 × ( 1 + ) $ 50 , 000 = $ 5 , 000 × ( 1 + r )12

10 000

, 5

$

000 ,

50

$ )

1 ( + r 12 = = ( 1 + r ) = 10112

2115

1 2115

1 1

=

r

About 21.15%

4.3 Compounding Periods

Compounding an investment m times a year for T

years provides for future value of wealth:

T m

m

r C

For example, if you invest $50 for 3 years at

12% compounded semi-annually, your investment will grow to

93 70

$ )

06 1 ( 50

$ 2

12

1 50

Effective Annual Interest Rates

A reasonable question to ask in the above example is

what is the effective annual rate of interest on that

investment?

The Effective Annual Interest Rate (EAR) is the annual rate that would give us the same end-of- investment wealth after 3 years:

93 70

$ )

06 1 ( 50

$

) 2

12

1 ( 50

FV

93 70

$ )

1 ( 50

$ × + EAR 3 =

Effective Annual Interest Rates (continued)

So, investing at 12.36% compounded annually is the same as investing at 12% compounded semiannually

93 70

$ )

1 ( 50

$ )

1 ( + EAR 3 =

1236

1 50

$

93 70

CompoundingNumber of times Effective

period compounded annual rate

Effective Annual Interest Rates (continued)

APR loan that is compounded monthly

rate of 1½ percent

rate of 19.56 percent

19561817

1 )

015

1

( 12

18

1

EARs and Compounding

• The Effective Annual Rate (EAR) is % The

“18% compounded semiannually” is the quoted

or stated rate, not the effective rate

be quoted on a loan agreement is equal to the rate per period multiplied by the number of periods

This rate is called the ( ).

What is the APR? What is the EAR?

Continuous Compounding

investment compounded continuously over many periods can be written as:

FV = C0×e rT

Where

r is the stated annual interest rate,

T is the number of periods over which the cash is

invested, and

e is a transcendental number approximately equal

The formula for the present value of a perpetuity is:



+ +

+ +

+ +

) 1

( )

1 ( )

1

C r

C r

C PV

r C

PV =

Perpetuity: Example

What is the value of a British consol that promises to pay £15 each year, every year until the sun turns into a red giant and burns the planet to a crisp?

The interest rate is 10-percent

× +

+

+

× +

+

) 1

(

) 1

( )

1 (

) 1

( )

1

g

C r

g

C r

C PV

g r

C PV

−

=

Growing Perpetuity: Example

The expected dividend next year is $1.30 and dividends are expected to grow at 5% forever

If the discount rate is 10%, what is the value of this promised dividend stream?

$ 05

10

.

30 1

The formula for the present value of an annuity is:

T

r

C r

C r

C r

C PV

) 1

( )

1 ( )

1 ( )

1 ( + + + 2 + + 3 + +

C PV

) 1

(

1 1

T C



Annuity: Example

If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?

12

$ )

12 07

1 (

1 1

12 /

07

• A A long time:

• What is the present value of a four-year annuity of

$100 per year that makes its first payment two years from today if the discount rate is 9%?

22 297

$

97 327

$ )

09 1 (

100

$ )

09 1 (

100

$ )

09 1 (

100

$ )

09 1 (

100

$ )

09 1 (

100

$

4 3

2 1

C r

C PV

) 1

(

) 1

( )

1 (

) 1

( )

1 (

1

+

× +

+ +

+

× +

r

C PV

) 1

(

1 1

PV of Growing Annuity

You are evaluating an income property that is providing

increasing rents Net rent is received at the end of each year

The first year's rent is expected to be $8,500 and rent is

expected to increase 7% each year Each payment occur at the

end of the year What is the present value of the estimated

income stream over the first 5 years if the discount rate is

12%?

0 1 2 3 4 5

500 ,

8

$

=

× ( 1 07 ) 500

, 8

$ $ 8 , 500 × ( 1 07 ) 2 =

095 ,

8

$

87 412 ,

8

$

77 141 ,

11

$

Growing Annuity

A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three- percent each year What is the present value at retirement if the discount rate is 10 percent?

$20,000

57 121 ,

265

$ 10

1

03

1 1

03 10

.

000 ,

PV of a delayed growing annuity

Your firm is about to make its initial public offering of stock and your job is to estimate the correct offering price Forecast dividends are as follows.

Dividends per share

$1.50 $1.65 $1.82 5% growth

thereafter

If investors demand a 10% return on investments of this risk level, what price will they be willing to pay?

PV of a delayed growing annuity

The first step is to draw a timeline.

The second step is to decide on what we know and what it is we are trying to find.

PV of a delayed growing annuity

Cash flow $1.50 $1.65 $1.82 dividend + P3

PV

of cash flow

$32.81

22 38

$ 05

10

05 1 82

$

22 38

$ 82

1

$ 65

1

$ 50

1

$

P

= $1.82 + $38.22

4.5 What Is a Firm Worth?

value of the firm’s cash flows

risk of those cash flows.

4.6 Summary and Conclusions

are introduced in this chapter

basis, but semi-annual, quarterly, monthly and even continuously compounded interest rate

arrangements exist

investment that pays $C for N periods is:

+

+ +

+ +

r

C r

C r

C C

NPV

1

0 2

0

) 1

( )

1 ( )

1 ( )

1

4.6 Summary and Conclusions (continued)

r

C

PV =

:Perpetuity

g r

C PV

−

=

:PerpetuityGrowing

C PV

)1

(

11

:Annuity

r

C PV

)1

(

11

:AnnuityGrowing

How do you get to Carnegie Hall?

yourself that you are a leotard purchase away from a triple back flip

time value of money problems and convince yourself that you can do them too

and flogging the keys until you can do these correctly and quickly

monthly payments over 10 years

tuition plan for your 8-year old daughter She will start college in exactly 10 years, with the first

tuition payment of $12,500 due at the start of the year Sophomore year tuition will be $15,000; junior year tuition $18,000, and senior year tuition

$22,000 How much money will you have to pay today to fully fund her tuition expenses? The

discount rate is 14%

you current car exactly 3 years ago for $25,000 and financed it at 7% APR for 60 months You need to estimate how much you owe on the loan to make sure that you can pay it off when you sell the old car

saving for a down-payment on a house You want

to save 20 percent of the purchase price and then borrow the rest from a bank

Houses that you like and can afford currently cost

$100,000 Real estate has been appreciating in price at 5 percent per year and you expect this trend to continue

to have a down payment saved five years from today?